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                <h1 id="_1">随机变量的数字特征</h1>
<h2 id="_2">数学期望</h2>
<p>设离散型随机变量X的分布律为<span class="arithmatex">\(P\{X=x_k\}=p_k,k=1,2,\cdots\)</span>，若级数<span class="arithmatex">\(\sum_{k=1}^\infty x_kp_k\)</span>绝对收敛，则称该级数的和为随机变量X的<strong>数学期望</strong>，记为E(X)，即</p>
<div class="arithmatex">\[E(X)=\sum_{k=1}^\infty x_kp_k\]</div>
<p>设连续性随机变量X的概率密度为f(x)，若积分<span class="arithmatex">\(\int_{-\infty}^\infty xf(x)dx\)</span>绝对收敛，则称该积分的值为随机变量X的<strong>数学期望</strong>，记为E(X)，即</p>
<div class="arithmatex">\[E(X)=\int_{-\infty}^\infty xf(x)dx\]</div>
<p>数学期望简称<strong>期望</strong>，又称为<strong>均值</strong></p>
<h3 id="_3">常见分布的数学期望：</h3>
<ol>
<li>泊松分布：<span class="arithmatex">\(若X\sim \pi(\lambda),即P=\{X=k\}=\frac{\lambda^ke^{-\lambda}}{k!}则E(X)=\lambda\)</span></li>
<li>连续分布：<span class="arithmatex">\(若X\sim U(a,b)，则E(X)＝\frac{a+b}{2}\)</span></li>
<li>指数分布：<span class="arithmatex">\(若X\sim E(\theta)，即f(x)=\frac{1}{\theta} e^{-x/\theta},x&gt;0或f(x)=0,x\leq 0，则E(X)=\theta\)</span></li>
<li>二项分布：np</li>
</ol>
<h3 id="_4">随机变量函数的数学期望</h3>
<p>设随机变量X的函数：Y=g(X)，g是连续函数
如果X是离散型随机变量，若<span class="arithmatex">\(\sum_{k=1}^\infty g(x_k)p_k\)</span>绝对收敛，则有<span class="arithmatex">\(<span class="arithmatex">\(E(Y)=E[g(X)]=\sum_{k=1}^\infty g(x_k)p_k\)</span>\)</span></p>
<p>如果X是连续型随机变量，若<span class="arithmatex">\(\int_{-\infty}^\infty g(x)f(x)dx\)</span>绝对收敛，则有</p>
<div class="arithmatex">\[E(Y)＝E[g(X)]=\int_{-\infty}^\infty g(x)f(x)dx\]</div>
<p>可以推广到两个或两个以上随机变量的函数的情况，例如：</p>
<div class="arithmatex">\[E(Z)=E[g(X,Y)]=\int_{-\infty}^\infty \int_{-\infty}^\infty g(x,y)f(x,y)dxdy\]</div>
<h3 id="_5">数学期望的性质</h3>
<ol>
<li>C是常数，则<span class="arithmatex">\(E(C)=C\)</span></li>
<li>X是随机变量，C是常数，则<span class="arithmatex">\(E(CX)=CE(X)\)</span></li>
<li>X和Y是随机变量，则<span class="arithmatex">\(E(X+Y)=E(X)+E(Y)\)</span>（可推广到任意有限个）</li>
<li>X和Y是<em>相互独立</em>的随机变量，则<span class="arithmatex">\(E(XY)=E(X)E(Y)\)</span>（可推广到任意有限个）</li>
</ol>
<h2 id="_6">方差</h2>
<p>设X是一个随机变量，若<span class="arithmatex">\(E\{[X-E(X)]^2\}\)</span>存在，则称他为X的<strong>方差</strong>，记为<span class="arithmatex">\(D(X)\)</span>或<span class="arithmatex">\(Var(X)\)</span>。应用上一节定理，可以得到<span class="arithmatex">\(D(X)=E(X^2)-[E(X)]^2\)</span>。为了统一单位，引入<span class="arithmatex">\(\sqrt{D(X)}\)</span>，记为<span class="arithmatex">\(\sigma(X)\)</span>，称为<strong>标准差</strong>或<strong>均方差</strong>。方差越小表明数据越集中。</p>
<h3 id="_7">标准化向量</h3>
<p>设X是随机变量，<span class="arithmatex">\(E(X)=\mu,D(X)=\sigma^2\)</span>，则称<span class="arithmatex">\(X^*=\frac{X-\mu}{\sigma}\)</span>为标准化向量，满足<span class="arithmatex">\(E(X^*)=0,D(X^*)=1\)</span></p>
<h3 id="_8">常见分布的方差</h3>
<ol>
<li>泊松分布：泊松分布的数学期望和方差都等于<span class="arithmatex">\(\lambda\)</span></li>
<li>连续分布：<span class="arithmatex">\(D(X)=\frac{(b-a)^2}{12}\)</span></li>
<li>指数分布：<span class="arithmatex">\(D(X)=\theta^2\)</span></li>
<li>二项分布：np(1-p)</li>
</ol>
<h3 id="_9">方差的性质</h3>
<ol>
<li>C是常数，则<span class="arithmatex">\(D(C)=0\)</span></li>
<li>X是随机变量，C是常数，则<span class="arithmatex">\(D(CX)=C^2D(X), D(X+C)=D(X)\)</span></li>
<li>X和Y是随机变量，则<span class="arithmatex">\(D(X+Y)=D(X)+D(Y)+2E\{(X-E(X))(Y-E(Y))\}=D(X)+D(Y)+2Cov(X,Y)\)</span>，若X和Y相互独立，则有<span class="arithmatex">\(D(X+Y)=D(X)+D(Y)\)</span>（可推广到任意有限个相互独立的随机变量之和）</li>
<li>D(X)=0的充要条件是X以概率1取常数E(X)，即<span class="arithmatex">\(P\{X=E(X)\}=1\)</span><blockquote>
<p>性质一和四的区别在于性质1是常数，性质4是随机变量</p>
</blockquote>
</li>
</ol>
<h3 id="_10">正态分布再研究</h3>
<p>对于<span class="arithmatex">\(X\sim N(\mu,\sigma^2)\)</span>，有<span class="arithmatex">\(E(X)=\mu,D(X)=\sigma^2\)</span>。
若<span class="arithmatex">\(X_i\sim N(\mu_i,\sigma_i^2),i=1,2,\cdots\)</span>且他们相互独立，则他们的线性组合</p>
<div class="arithmatex">\[C_1X_1+C_2X_2+\cdots+C_nX_n\sim N(\sum_{i=1}^n C_i\mu_i,\sum_{i=1}^n C_i^2\sigma_i^2)\]</div>
<h3 id="_11">切比雪夫不等式</h3>
<p>设随机变量X具有数学期望<span class="arithmatex">\(E(X)=\mu\)</span>，方差<span class="arithmatex">\(D(X)=\sigma^2\)</span>，则对于任意正数<span class="arithmatex">\(\epsilon\)</span>，不等式</p>
<div class="arithmatex">\[P\{|X-\mu|\geq \epsilon\}\leq\frac{\sigma^2}{\epsilon^2}\]</div>
<p>成立。这被称为切比雪夫不等式。
也可以被写成：</p>
<div class="arithmatex">\[P\{|X-\mu|&lt; \epsilon\}\geq1-\frac{\sigma^2}{\epsilon^2}\]</div>
<h2 id="_12">协方差和相关系数</h2>
<p>量<span class="arithmatex">\(E\{[X-E(X)][Y-E(Y)]\}\)</span>称为随机变量X与Y的<strong>协方差</strong>，记为Cov(X,Y)，即</p>
<div class="arithmatex">\[Cov(X,Y)=E\{[X-E(X)][Y-E(Y)]\}\]</div>
<p>而<span class="arithmatex">\(\rho_{XY}=\frac{Cov(X,Y)}{\sqrt{D(X)}\sqrt{(D(Y)}}\)</span>称为随机变量X与Y的<strong>相关系数</strong>
有定义即可得到协方差性质：</p>
<div class="arithmatex">\[Cov(X,Y)=Cov(Y,X),Cov(X,X)=D(X)\]</div>
<p>展开协方差定义式得到</p>
<div class="arithmatex">\[Cov(X,Y)=E(XY)-E(X)E(Y)\]</div>
<h3 id="_13">协方差性质</h3>
<ol>
<li><span class="arithmatex">\(Cov(aX,bY)=abCov(X,Y)\)</span>，a,b是常数</li>
<li><span class="arithmatex">\(Cov(X_1+X_2,Y)=Cov(X_1,Y)+Cov(X_2,Y)\)</span></li>
</ol>
<h3 id="_14">相关系数性质</h3>
<p>引入均方误差<span class="arithmatex">\(e=E\{[Y-(a+bX)]^2\})\)</span>来衡量<span class="arithmatex">\(a+bX\)</span>近似表达Y的好坏程度，当e取到最小时有<span class="arithmatex">\(b_0=\frac{Cov(X,Y)}{D(X)}\)</span>,<span class="arithmatex">\(a_0=E(Y)-b_0E(X)\)</span>,<span class="arithmatex">\(min\{e\}=(1-\rho_{XY}^2)D(Y)\)</span>
由此得到相关系数的性质</p>
<ol>
<li><span class="arithmatex">\(|\rho_{XY}|\leq 1\)</span></li>
<li><span class="arithmatex">\(|\rho_{XY}|= 1\)</span>当且仅当存在常数a,b使得<span class="arithmatex">\(P\{Y=a+bX\}=1\)</span>
并得到相关系数的象征：相关系数越大均方误差越小，X和Y的线性关系越紧密；当<span class="arithmatex">\(|\rho_{XY}|=1\)</span>时，X和Y之间以概率1存在着线性关系，当<span class="arithmatex">\(|\rho_{XY}|=0\)</span>时，称X和Y <strong>不相关</strong>。</li>
</ol>
<p>如果X和Y的相关系数存在，X和Y独立可推出X和Y不相关，不能反推。</p>
<p>二维正态分布的参数<span class="arithmatex">\(\rho\)</span>就是X和Y的相关系数，也就是说二维正态分布可由X和Y的期望、方差和相关系数完全确定；可推出对于二维正态分布，X和Y独立和X和Y不相关（<span class="arithmatex">\(\rho=0\)</span>）是等价的。</p>
<h2 id="_15">矩、协方差矩阵</h2>
<p>若X和Y是随机变量：
若<span class="arithmatex">\(E(X^k),k=1,2,\cdots\)</span>存在，则称它为X的<strong>k阶原点矩</strong>，简称<strong>k阶矩</strong>
若<span class="arithmatex">\(E\{[X-E(X)]^k\},k=1,2,\cdots\)</span>存在，则称它为X的<strong>k阶中心矩</strong>
若<span class="arithmatex">\(E(X^kY^l)\)</span>存在，则称它为X和Y的<strong>k+l阶混合矩</strong>
若<span class="arithmatex">\(E\{[X-E(X)]^k[Y-E(Y)]^l\}\)</span>存在，则称它为X和Y的<strong>k+l阶混合中心矩</strong></p>
<p>设<span class="arithmatex">\((X_1,X_2,\cdots,X_n)\)</span>为n维随机变量，记<span class="arithmatex">\(c_{ij}=Cov(X_i,X_j)\)</span>，则n维矩阵<span class="arithmatex">\(\textbf{C}=[c_{ij}]\)</span>称为该n维随机变量的<strong>协方差矩阵</strong>。</p>
<p>对于n维随机变量<span class="arithmatex">\((X_1,X_2,\cdots,X_n)\)</span>，记：</p>
<div class="arithmatex">\[\textbf{X}=\left[\begin{aligned}x_1\\x_2\\\vdots\\ x_n\end{aligned}\right],\boldsymbol{\mu}=\left[\begin{aligned}E(x_1)\\E(x_2)\\\vdots\\ E(x_n)\end{aligned}\right]\]</div>
<p>则定义n维正态随机变量<span class="arithmatex">\((X_1,X_2,\cdots,X_n)\)</span>的概率密度为</p>
<div class="arithmatex">\[f(x_1,x_2,\cdots,x_n)=\frac{1}{(2\pi)^{n/2}(det\textbf{C})^{1/2}}exp\left\{-\frac{1}{2}(\textbf{X}-\boldsymbol{\mu})^T\textbf{C}^{-1}(\textbf{X}-\boldsymbol{\mu})\right\}\]</div>
<p>其具有以下性质：</p>
<ol>
<li>
<p>n维正态随机变量<span class="arithmatex">\((X_1,X_2,\cdots,X_n)\)</span>中的每个分量<span class="arithmatex">\(X_i,i=1,2,\cdots,n\)</span>都是正态随机变量；反之，若<span class="arithmatex">\(X_1,X_2,\cdots,X_n\)</span>都是正态随机变量，且相互独立，则<span class="arithmatex">\((X_1,X_2,\cdots,X_n)\)</span>是n维正态随机变量</p>
</li>
<li>
<p>n维随机变量<span class="arithmatex">\((X_1,X_2,\cdots,X_n)\)</span>服从n维正态分布的充要条件是<span class="arithmatex">\(X_1,X_2,\cdots,X_n\)</span>的任意线性组合<span class="arithmatex">\(l_1X_1+l_2X_2+\cdots+l_nX_n\)</span>服从一维正态分布（其中<span class="arithmatex">\(l_1,l_2,\cdots,l_n\)</span>不全为零）</p>
</li>
<li>
<p>若<span class="arithmatex">\((X_1,X_2,\cdots,X_n)\)</span>服从n维正态分布，设<span class="arithmatex">\((Y_1,Y_2,\cdots,Y_k)\)</span>是<span class="arithmatex">\(X_j(j=1,2,\cdots,n)\)</span>的线性函数，则<span class="arithmatex">\((Y_1,Y_2,\cdots,Y_k)\)</span>也服从多维正态分布。这被称为正态变量的<em>线性不变性</em></p>
</li>
<li>
<p>设<span class="arithmatex">\((X_1,X_2,\cdots,X_n)\)</span>服从n维正态分布，则“<span class="arithmatex">\(X_1,X_2,\cdots,X_n\)</span>相互独立”与“<span class="arithmatex">\(X_1,X_2,\cdots,X_n\)</span>两两不相关”是等价的。</p>
</li>
</ol>
              
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